Method for the blind wideband localization of one or more transmitters from a carrier that is passing by

ABSTRACT

A method for localizing one or more sources, the source or sources being in motion relative to a network of sensors, comprises a step for the separation of the sources in order to identify the direction vectors associated with the response of the sensors to a source at a given incidence. The method comprises the following steps of associating the direction vectors a 1p(1)m  . . . a Kp(K)m  obtained for the m th  transmitter and, respectively, for the instants t 1  . . . t K  and for the wavelengths λ p(1)  . . . , λ p(k) , and localizing the m th  transmitter from the components of the vectors a 1p(1)m  . . . a Kp(K)m  measured with different wavelengths.

RELATED APPLICATIONS

The present application is based on, and claims priority from, France Application Number 04 03295, filed Mar. 30, 2004, the disclosure of which is hereby incorporated by reference herein in its entirety.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention relates to a method for localizing a plurality of ground transmitters in a broadband context from the passing by of carrier without a priori knowledge on the signals sent. The carrier may be an aircraft, a helicopter, a ship, etc.

The method is implemented, for example iteratively, during the passing by of the carrier.

2. Description of the Prior Art

The prior art describes different methods to localize one or more transmitters from a passing carrier.

FIG. 1 illustrates an example of airborne localization. The transmitter 1 is in the position (x₀,y₀,z₀) and the carrier 2 at the instant t_(k) is at the position (x_(k),y_(k),z_(k)) and perceives the transmitter at the incidence (θ(t_(k,)x₀,y₀,z₀),Δ(t_(k),x₀,y₀,z₀)). The angles θ(t,x₀,y₀,z₀) and Δ(t,x₀,y₀,z₀) evolve in time and depend on the position of the transmitter as well as the trajectory of the carrier. The angles θ(t,x₀,y₀,z₀) and Δ(t,x₀,y₀,z₀) are identified as can be seen in FIG. 2 relative to a network 3 of N antennas capable of being fixed under the carrier.

There are many classes of techniques used to determine the position (x_(m),y_(m),z_(m)) of the transmitters. These techniques differ especially in the parameters instantaneously estimated at the network of sensors. Thus, localising techniques can be classified under the following categories: use in direction-finding, use of the phase difference between two distant sensors, use of the measurement of the carrier frequency of the transmitter, use of the propagation times.

The patent application FR 03/13128 by the present applicant describes a method for localising one or more transmitters from a passing carrier where the direction vectors are measured in the same frequency channel and are therefore all at the same wavelength.

The method according to the invention is aimed especially at achieving a direct estimation of the positions (x_(m),y_(m),z_(m)) of each of the transmitters from a blind identification of the direction vectors of the transmitters at various instants t_(k) and various wavelengths λ_(k).

Parametrical analysis will have the additional function of separating the different transmitters at each wavelength-instant pair (t_(k), λ_(p(k))). The parameters of the vectors coming from the different pairs (t_(k), λ_(p(k))) are then associated so that, finally, a localisation is performed on each of the transmitters.

SUMMARY OF THE INVENTION

The invention relates to a method for localizing one or more sources, said source or sources being in motion relative to a network of sensors, the method comprising a step for the separation of the sources in order to identify the direction vectors associated with the response of the sensors to a source at a given incidence. It is characterised in that it comprises at least the following steps:

-   -   associating the direction vectors a_(1p(1)m) . . . a_(Kp(K)m)         obtained for the m^(th) transmitter and, respectively, for the         instants t₁ . . . t_(K) and for the wavelengths λ_(p(1)) . . . ,         λ_(p(k)),     -   localizing the m^(th) transmitter from the components of the         vectors a_(1p(1)m) . . . a_(Kp(K)m) measured for the different         wavelengths.

The wideband method according to the invention offers notably the following advantages:

-   -   the localising of the transmitter is done by a direct method         which maximises a single criterion as a function of the (x,y,z)         coordinates of the transmitter,     -   it makes it possible to achieve an association of the direction         vectors of the sources in the time-frequency space, making it         possible especially to take EVF (Evasion de Fréquence=Frequency         Evasion) and TDMA-FDMA (Time Division Multiple Access and         Frequency Division Multiple Access) signals into account,     -   it can be implemented on calibrated networks or with amplitude         diversity antennas such as co-localised antennas: namely         antennas in a network with dipoles having a same phase centre         and different orientations.

BRIEF DESCRIPTION OF THE DRAWINGS

Other features and advantages of the present information shall appear more clearly from the following description of a detailed example, given by way of an illustration that in no way restricts the scope of the invention, and from the appended figures, of which:

FIG. 1 exemplifies a localisation, by an aircraft equipped with a network of antennas, of a transmitter having a position (x₀, y₀, z₀) on the ground,

FIG. 2 shows a network of five antennas and the angles of incidence of a transmitter,

FIG. 3 is a graph showing the general operations of the method in the presence of M wideband transmitters.

DETAILED DESCRIPTION OF THE DRAWINGS

For a clear understanding of the method according to the invention, the following example is given by way of illustration that in no way restricts the scope of the invention, for a system as described in FIG. 1 comprising an aircraft 2 equipped with a reception device comprising, for example, a network of N sensors (FIG. 2), and a transmitter 1 to be localised.

Before explaining the steps of the method according to the invention, the model used is defined.

Modelling

In the presence of M transmitters, the aircraft receives the vector x(t,p) at the instant t at output of the N sensors of the network and of the p^(th) channel having a wavelength λp.

$\begin{matrix} {{x\left( {{t + t_{k}},p} \right)} = {\begin{bmatrix} {x_{1}\left( {{t + t_{k}},p} \right)} \\ M \\ {x_{N}\left( {{t + t_{k}},p} \right)} \end{bmatrix}\mspace{115mu} = {{{\sum\limits_{m = 1}^{M}{{a\left( {\theta_{km},\Delta_{km},\lambda_{p}} \right)}{s_{m}\left( {t + t_{k}} \right)}}} + {b\left( {t + t_{k}} \right)}}\mspace{115mu} = {{{A_{kp}{s\left( {{t + t_{k}},p} \right)}} + {{b\left( {{t + t_{k}},p} \right)}\mspace{14mu}{for}\mspace{14mu}{t}}} < {\Delta\;{t/2}}}}}} & (1) \end{matrix}$

-   -   where     -   b(t) is the noise vector assumed to be Gaussian,     -   a(θ,Δ,λ) is the response of the network of sensors to a source         with an incidence (θ,Δ) and a wavelength λ,     -   A_(kp)=[a(θ_(k1),Δ_(k1),λ_(p)) . . . a(θ_(kM),Δ_(kM),λ_(p))]         corresponds to the mixing matrix, s(t)=[s₁(t) . . .         s_(M)(t)]^(T) corresponds to the direction vector,     -   θ_(km)=θ(t_(k),x_(m),y_(m),z_(m)) and         Δ_(km)=Δ(t_(k),x_(m),y_(m),z_(m)).     -   x_(n)(t, p) is the signal received on the n^(th) sensor of the         carrier at output of the p^(th) frequency channel associated         with the wavelength λ_(p).

In this model, the mixing matrix A_(kp) depends on the instant t_(k) of observation as well as on the wavelength λ_(p).

The above model shows that the direction vector: a _(kpm) =a(θ_(km),Δ_(km),λ_(p))=a(t _(k) ,λ _(p) ,x _(m) ,y _(m) ,z _(m)) of the m^(th) transmitter  (2) at the instant t_(k) is a known function of (t_(k),λ_(p)) and of the position of the transmitter (x_(m),y_(m),z_(m)). The method according to the invention comprises, for example, the following steps summarised in FIG. 3

-   -   the parametrical estimation (PE) and the separation of the         transmitters

(ST) at the instant t_(k) and wavelength λ_(p) for example in identifying the vectors a_(kpm) for (1≦m≦M).

This first step is effected by techniques of source separation and identification described, for example, in the references [2] [3],

-   -   the association of the parameters for the m^(th) transmitter,         for example in associating the vectors a_(1p(1)m) up to         a_(Kp(K)m) obtained at the respective pairs (instants,         wavelength) (t₁λ_(p(1))) . . . (t_(K),λ_(p(K))).         The direction vectors a_(kpm) are considered in a (time,         wavelength) space or, again, in a (time, frequency) space, the         frequency being inversely proportional to the wavelength,     -   The localizing of the m^(th) transmitter from the vectors         a_(1p(1)m) up to a_(Kp(K)m) associated, LOC-WB.         Step of Association and Tracking of the Wideband Transmitters

In the presence of M sources or transmitters, after the source separation step where the direction vectors are identified but not associated, the method gives, for the instant/wavelength pair (t_(k), λ_(p)), the M_(k) signatures a_(kpm) for (1≦m≦M_(k)), signature or vector associated with a source.

At the instant t_(k′) and at the wavelength λ_(p′), the source separation step gives the M_(k′) vectors b_(i) for (1≦i≦M_(k′)). The purpose of the tracking of the transmitters is especially to determine, for the m^(th) transmitter, the index i(m) which minimizes the difference between the vectors a_(kpm) and b_(i(m)). In this case, it will be deduced therefrom that a_(k′p′m)=b_(i(m)).

To make the association of the parameters for the m^(th) transmitter,

a criterion of distance is defined between two vectors u and v giving:

$\begin{matrix} {{d\left( {u,v} \right)} = {1 - \frac{{{u^{H}v}}^{2}}{\left( {u^{H}u} \right)\left( {v^{H}v} \right)}}} & (3) \end{matrix}$ H corresponds to the transpose of the vectors u or v Thus, the index i(m) sought verifies:

$\begin{matrix} {{d\left( {a_{kpm},b_{i{(m)}}} \right)} = {\min\limits_{1 \leq i \leq M_{k^{\prime}}}\left\lbrack {d\left( {a_{kpm},b_{i}} \right)} \right\rbrack}} & (4) \end{matrix}$ In this association, we consider a two-dimensional function associated with the m^(th) transmitter defined by: {circumflex over (β)}_(m)(t _(k),λ_(p))=d(a _(kp(k)m) , a _(00m))  (5)

In the course of the association, there is obtained, by interpolation of the {circumflex over (β)}_(m)(t_(k),λ_(p)) values for each transmitter indexed by m, a function β_(m)(t,λ) for 1≦m≦M. This function has the role especially of eliminating the pairs (t_(k),λ_(p)) such that β_(m)(t_(k),λ_(p)) and {circumflex over (β)}_(m)(t_(k),λ_(p)) are very different: |{circumflex over (β)}_(m)(t_(k),λ_(p))−β_(m)(t_(k),λ_(p))|>η. Thus the aberrant instants which may be associated with other transmitters are eliminated.

Since the β_(m)(t,λ) brings into play the distance d(u,v) between the vectors u and v, it is said that u and v are close when: d(u, v)<η  (6) The value of the threshold η is chosen for example as a function of the following error model: u=v+e  (7) where e is a random variable. When the direction vectors are estimated on a duration of T samples, the law of the variable e may be approached by a Gaussian mean standard deviation law σ=1/√{square root over (T)}. Thus, the distance d(u, v) is proportional to a chi-2 law with (N−1) degrees of freedom (N:length of the vectors u and v). The ratio between the random variable d(u, v) and the chi-2 law is equal to σ/N. With the law of e being known, it is possible to determine the threshold η with a certain probability of false alarms. In the steps of the association of the method, a distance d_(ij) is defined in the time-wavelength space between the pairs (t_(i),λ_(p(i))) and (t_(j),λ_(p(j))): d _(ij)=√{square root over ((t _(i) −t _(j))²+(λ_(p(i))−λ_(p(j)))²)}{square root over ((t _(i) −t _(j))²+(λ_(p(i))−λ_(p(j)))²)}{square root over ((t _(i) −t _(j))²+(λ_(p(i))−λ_(p(j)))²)}  (8) In considering that, for each pair, (t_(k),λ_(p(k))), M_(k) vectors a_(kp(k)j) (1<j<M_(k)) have been identified, the steps of this association for K pairs (t_(k),λ_(p(k))) are given here below. The steps of association for K instants t_(k) and λ_(p) wavelengths are, for example, the following:

-   -   Step AS1 Initialization of the process at k=0, m=1 and M=1. The         initial number of transmitters is determined, for example, by         means of a test to detect the number of sources at the instant         t0, this being a test known to those skilled in the art.         -   For all the triplets (t_(k),λ_(p(k)),j) the initialization             of a flag called flag_(kj) with flag_(kj)=0: flag_(kj)=0             indicates that the j^(th) direction vector obtained at             (t_(k),λ_(p(k))) is not associated with any family of             direction vectors.     -   Step AS2 Search for an index j and a pair (t_(k),λ_(p(k))) such         that flag_(kj)=0, which expresses the fact that the association         with a family of direction vectors is not achieved.     -   Step AS3 For this 1^(st) triplet (t_(k),λ_(p(k)),j), flag_(kj)=1         is effected and link_(k′i) is initialized at link_(k′i)=0 for         k′≠k and i′≠j and ind_(m)={k} and Φ_(m)={a_(kp(k)j)}, where         Φ_(m) is the set of vectors associated with the m^(th) familly         or m^(th) transmitter, ind_(m) the set of the indices k of the         pairs (t_(k),λ_(p(k))) associated with the same transmitter,         link_(ki) is a flag that indicates whether it has performed a         test of association of the vector a_(kp(k)i) of of the triplet         (t_(k),λ_(p(k)),i) with the m^(th) family of direction vectors:         link_(ki)=0 indicates that the association test has not been         performed.     -   Step AS4 Determining the pair (t_(k′),λ_(p(k′))) minimizing the         distance d_(kk′) of the relationship (8) with (t_(k),λ_(p(k)))         such that kεind_(m) in the time-frequency space and in which         there exists at least one vector b_(i)=a_(k′p(k′)i) such that         flag_(k′i)=0 and link_(k′i)=0; the vector has never been         associated with a family on the one hand and there has been no         test of association conducted with the m^(th) family on the         other hand. A search is made in a set of data resulting from the         separation of the sources.     -   Step AS5 By using the relationship (4) defined here above, we         determine the index i(m) minimizing the difference between the         vectors a_(kp(k)m) such that kεind_(m) and the vectors b_(i)         identified with the instant-wavelength pairs (t_(k′),λ_(p(k′)))         for (1≦i≦M_(k′)) and flag_(k′i)=0 and link_(k′i)=0.     -   Step AS6 do link_(k′i(m))=1: the test of the association with         m^(th) family has been performed.     -   Step AS7 If d(a_(kp(k)m), b_(i(m)))≦η relationship (6) and         |t_(k)−t_(k′)|<Δt_(max) and |λ_(p(k))−λ_(p(k′))|<Δλ_(max) then         perform Φ_(m)={Φ_(m)b_(i(m))}, ind_(m)={ind_(m)k′},         flag_(k′i(m))=1: The vector is associated with the m^(th)         family.     -   Step AS8 Return to the step AS4 if there is at least one doublet         (t_(k′),λ_(p(k′))) and one index i such that the flag         link_(k′i)=0 and flag_(k′i)=0.     -   Step AS9 In writing K(m)=cardinal(Φ_(m)), we obtain the family         of vectors Φ_(m)={a_(1p(1)m) . . . a_(K(m),p(K(m)),m)}         associated with the source indexed by m.     -   For each vector a_(kp(k)m) the estimate {circumflex over         (β)}_(m)(t_(k),λ_(p)) of (5) is associated and then a polynomial         interpolation is made of the {circumflex over         (β)}_(m)(t_(k),λ_(p)) to obtain the two-dimensional interpolated         function β_(m)(t,λ).     -   Step AS10 From the family of vectors Φ_(m)={a_(1p(1),m) . . .         a_(K(M),p(K(M)),m)}, extract the J instants t_(i) εind_(j)         ⊂ind_(m) such as the coefficients         |β_(m)(t_(i),λ_(p(i)))−{circumflex over         (β)}_(m)(t_(i),λ_(p(i)))|<η (threshold value) such that         β_(m)(t_(i),λ_(p(i))) is not an aberrant point of the function         β_(m)(t,λ)     -   It is said that there is an aberrant point when the divergence         in modulus between the point {circumflex over         (β)}_(m)(t_(i),λ_(p(i))) and an interpolation of the function         β_(m)(t,λ) does not exceed a threshold η.     -   After this sorting out, the new family of pairs is         Φ_(m)={a_(k,p(k),m)/k εind_(j)}, ind_(m)=ind_(j) and K(M)=J,         M←M+1 and m←M.     -   Step AS11 Return to the step AS3 if there is at least one         triplet (t_(k),λ_(p(k)),j) such that flag_(kj)=0.     -   Step AS12 M←M−1.         After the step AS10, we are in possession of the family of         vectors Φ_(m)={a_(1p(1),m) . . . a_(K(M),p(K(M)),m)} having no         aberrant points. Each vector has an associated function         β_(m)(t,λ) whose role especially is to eliminate the aberrant         points which do not belong to a zone of uncertainty given by η         (see equations (5)(6)(7)).     -   The steps of the method described here above have notably the         following advantages:     -   The number M of transmitters is determined automatically,         The following are determined for each transmitter:         -   The class of vectors Φ_(m)={a_(1p(1),m) . . .             a_(K(m),p(K(m)),m)},         -   The number K(m) of direction vectors,         -   A set ind of indices indicating the pairs (t_(k),λ_(p(k)))             associated with the vectors of the set Φ_(m).     -   Managing the case of the appearance and disappearance of a         transmitter,     -   Associating the transmitters in the time-wavelength space (t,λ).

The method of association described here above by way of an illustration that in no way restricts the scope of the invention is based on a criterion of distance of the direction vectors. Without departing from the scope of the invention, it is possible to add other criteria to it such as:

-   -   the signal level received in the channel considered (criterion         of correlation on the level),     -   the instant of the start of transmission (front) or detection of         a periodic marker (reference sequence), enabling the use of         synchronisation criteria in the case of time-synchronised EVF         signals (steady levels, TDMA, bursts, etc.),     -   characteristics related to the technical analysis of the signal         (waveform, modulation parameters . . . ),     -   etc.

The following step is that of localising the transmitters.

The Wideband Localisation of a Transmitter

The goal of the method especially is to determine the position of the m^(th) transmitter from the components of the vectors a_(1p(1)m) up to a_(Kp(K)m) measured with different wavelengths.

These vectors a_(kp(k)m) have the particular feature of depending on the instant t_(k), the wavelength λ_(p(k)) and the position (x_(m),y_(m),z_(m)) of the transmitter.

For example, for a network formed by N=2 sensors spaced out by a distance of d in the axis of the carrier, the direction vector verifies a_(kp(k)m).

$\begin{matrix} {a_{{{kp}{(k)}}m} = {\begin{bmatrix} 1 \\ {\exp\left( {{j2}\;\pi\;\frac{d}{\lambda_{p{(k)}}}{\cos\left( {\theta\left( {t_{k},x_{m},y_{m},z_{m}} \right)} \right)}\cos} \right.} \\ \left. \left( {\Delta\left( {t_{k},x_{m},y_{m},z_{m}} \right)} \right) \right) \end{bmatrix}\mspace{65mu} = {a\left( {t_{k},\lambda_{p{(k)}},x_{m},y_{m},z_{m}} \right)}}} & (9) \end{matrix}$

According to FIG. 1, the incidence (θ(t_(k),x_(m),y_(m),z_(m)),Δ(t_(k),x_(m),y_(m),z_(m))) may be computed directly from the position (x_(k),y_(k),z_(k)) of the carrier at the instant t_(k) and the position (x_(m),y_(m),z_(m)) of the transmitter.

The method will, for example, build a vector b_(kp(k)m) from components of the vector a_(kp(k)m). The vector b_(kp(k)m) may be a vector with a dimension (N−1)×1 in choosing the reference sensor in n=i:

$\begin{matrix} {b_{{{kp}{(k)}}m} = {\begin{bmatrix} {{a_{{{kp}{(k)}}m}(1)}/{a_{{{kp}{(k)}}m}(i)}} \\ M \\ {{a_{{{kp}{(k)}}m}\left( {i - 1} \right)}/{a_{{{kp}{(k)}}m}(i)}} \\ {{a_{{{kp}{(k)}}m}\left( {i + 1} \right)}/{a_{{{kp}{(k)}}m}(i)}} \\ M \\ {{a_{{{kp}{(k)}}m}(N)}/{a_{{{kp}{(k)}}m}(i)}} \end{bmatrix}\mspace{65mu} = {b\left( {t_{k},\lambda_{p{(k)}},x_{m},y_{m},z_{m}} \right)}}} & (10) \end{matrix}$ where a_(kp(k)m)(n) is the i^(th) component of a_(kp(k)m). Thus, in the example of the equation (9) and in fixing i=1 we get:

$\begin{matrix} {b_{{{kp}{(k)}}m} = \begin{bmatrix} {\exp\left( {{j2}\;\pi\;\frac{d}{\lambda_{p{(k)}}}{\cos\left( {\theta\left( {t_{k},x_{m},y_{m},z_{m}} \right)} \right)}\cos} \right.} \\ \left. \left( {\Delta\left( {t_{k},x_{m},y_{m},z_{m}} \right)} \right) \right) \end{bmatrix}} & (11) \end{matrix}$

It being known that the direction vectors a_(kp(k)m) are estimated with a certain error e_(kp(k)m) such that a_(kp(k)m)=a(t_(k),λ_(p(k)),x_(m),y_(m),z_(m))+e_(kp(k)m). The same is true for the transformed vector b_(kp(k)m) of (10) at the first order when ∥e_(kp(k)m)∥<<1. b _(kp(k)m) =b(t _(k),λ_(p(k)) ,x _(m) ,y _(m) ,z _(m))+w _(km)  (12)

The family of localizing techniques mentioned in the prior art, using the phase shift between two sensors, requires knowledge of the phase of the vector b_(kp(k)m). It being known that the vector a_(kp(k)m) is a function of the position (x_(m),y_(m),z_(m)) of the transmitter, the same is true for the vector b_(kp(k)m).

Under these conditions, the localisation method consists, for example, in maximizing the following criterion of standardized vector correlation L_(K)(x,y,z) in the position space (x,y,z) of a transmitter.

$\begin{matrix} {{{{{L_{K}\left( {x,y,z} \right)} = {\frac{{{b_{K}^{H}{v_{K}\left( {x,y,z} \right)}}}^{2}}{\left( {b_{K}^{H}b_{K}} \right)\left( {{v_{K}\left( {x,y,z} \right)}^{H}{v_{K}\left( {x,y,z} \right)}} \right)}\mspace{14mu}{with}}}b_{K} = {\begin{bmatrix} b_{1m} \\ M \\ b_{Km} \end{bmatrix}\mspace{31mu} = {{v_{K}\left( {x_{m},y_{m},z_{m}} \right)} + w_{K}}}},{{v_{K}\left( {x,y,z} \right)}\mspace{31mu} = \begin{bmatrix} {b\left( {t_{1},\lambda_{p{(1)}},x,y,z} \right)} \\ M \\ {b\left( {t_{K},\lambda_{p{(K)}},x,y,z} \right)} \end{bmatrix}}}{and}{w_{K} = \begin{bmatrix} w_{1m} \\ M \\ w_{Km} \end{bmatrix}}} & (13) \end{matrix}$ The noise vector w_(K) has the matrix of covariance R=E[w_(K) w_(K) ^(H)]. Assuming that it is possible to know this matrix R, the criterion may be envisaged with a whitening technique. In these conditions, the following criterion L_(K)′(x,y,z) is obtained

$\begin{matrix} {{{L_{K}^{\prime}\left( {x,y,z} \right)} = \frac{{{b_{K}^{H}R^{- 1}{v_{K}\left( {x,y,z} \right)}}}^{2}}{\left( {b_{K}^{H}R^{- 1}b_{K}} \right)\left( {{v_{K}\left( {x,y,z} \right)}^{H}R^{- 1}{v_{K}\left( {x,y,z} \right)}} \right)}}{{{with}\mspace{14mu} R} = {E\left\lbrack {w_{K}w_{K}^{H}} \right\rbrack}}} & (14) \end{matrix}$

The vector V_(K) (x,y,z) depends on the K wavelengths λ_(p(1)) up to λ_(p(K)). This is why it is said that the method achieves a broadband localisation.

The criteria L_(K)(x,y,z) and L_(K)′(x,y,z) have the advantage of enabling the implementation of a localization technique in the presence of a network of sensors calibrated in space (θ,Δ) at various wavelengths λ.

Given that, at the instant t_(k), we know the analytic relationship linking the incidence (θ(t_(k),x,y,z), Δ(t_(k),x,y,z)) of the transmitter with its position (x,y,z), it is then possible, from the incidence (θ(t_(k),x,y,z), Δ(t_(k),x,y,z)), to deduce the vector a(t_(k),λ_(p(k)),x_(m),y_(m),z_(m))=a(θ(t_(k),x,y,z) , Δ(t_(k),x,y,z),λ_(p(k))) in making an interpolation of the calibration table in the 3D space (θ,Δ,λ).

In an airborne context, the knowledge of the altitude h of the aircraft reduces the computation of the criterion in the search space (x,y), assuming z=h.

In the example of the equations (9) and (11) the vector v_(K) (x,y,z) is written as follows:

$\begin{matrix} {{v_{K}\left( {x,y,z} \right)} = \begin{bmatrix} {\exp\left( {{j2}\;\pi\;\frac{d}{\lambda_{p{(1)}}}{\cos\left( {\theta\left( {t_{1},x,y,z} \right)} \right)}{\cos\left( {\Delta\left( {t_{1},x,y,z} \right)} \right)}} \right)} \\ M \\ {\exp\left( {{j2}\;\pi\;\frac{d}{\lambda_{p{(K)}}}{\cos\left( {\theta\left( {t_{K},x,y,z} \right)} \right)}{\cos\left( {\Delta\left( {t_{K},x,y,z} \right)} \right)}} \right)} \end{bmatrix}} & (15) \end{matrix}$

In this method, it is possible to consider initialising the algorithm at K=K₀ and then recursively computing the criterion L_(K)(x,y,z).

Thus, L_(K)(x,y,z) is computed recursively as follows:

$\begin{matrix} {{{L_{K + 1}\left( {x,y,z} \right)} = {\frac{{{\alpha_{K + 1}\left( {x,y,z} \right)}}^{2}}{\beta_{K + 1}{\gamma_{K + 1}\left( {x,y,z} \right)}}\mspace{14mu}{where}}}{{\alpha_{K + 1}\left( {x,y,z} \right)} = {{\alpha_{K}\left( {x,y,z} \right)} + \mspace{169mu}{b_{K + {1{p{({K + 1})}}m}}^{H}{b\left( {{t_{{K + 1},}\lambda_{p{({K + 1})}}},x,y,z} \right)}}}}{{\gamma_{K + 1}\left( {x,y,z} \right)} = {{\gamma_{K}\left( {x,y,z} \right)} + {b\left( {t_{K + 1},\lambda_{p{({K + 1})}},x,y,z} \right)}^{H}}}\mspace{169mu}{b\left( {t_{K + 1},\lambda_{p{({K + 1})}},x,y,z} \right)}{\beta_{K + 1} = {\beta_{K} + {b_{K + {1{p{({K + 1})}}m}}^{H}b_{K + {1{p{({K + 1})}}m}}}}}} & (16) \end{matrix}$

When the vectors b(t_(K+1),λ_(p(K+1)),x,y,z) and b_(kp(k)m) are constant standards equal to ρ the relationship of recurrence of the equation (16) becomes:

$\begin{matrix} {{{L_{K + 1}\left( {x,y,z} \right)} = \frac{{{\alpha_{K + 1}\left( {x,y,z} \right)}}^{2}}{{\beta^{2}\left( {K + 1} \right)}^{2}}}{Where}\text{}{{\alpha_{K + 1}\left( {x,y,z} \right)} = {{\alpha_{K}\left( {x,y,z} \right)} + \mspace{169mu}{b_{K + {1{p{({K + 1})}}m}}^{H}{b\left( {t_{K + 1},\lambda_{p{({K + 1})}},x,y,z} \right)}}}}} & (17) \end{matrix}$

The method has been described up to this point in assuming that the transmitters have fixed positions. It can easily be extended to the case of moving targets with a speed vector (v_(xm),v_(ym),v_(zm)) for which there is a model of evolution as described in the patent application FR 03/13128.

The method according to the invention can be applied to a very large number of measurements. In this case, the value of K is diminished in order to reduce the numerical complexity of the computations.

The method provides, for example, for the performance of the following processing operations on the elementary measurements:

-   -   decimation of the pairs (t_(k), λ_(p(k))),     -   filtering (smoothing of the measurements which are the direction         vectors) and sub-sampling,     -   merging on a defined duration (extraction by association of         direction vectors to produce a measurement of synthesis).

BIBLIOGRAPHY

-   1—R O. SCHMIDT—November 1981—A signal subspace approach to multiple     emitter location and spectral estimation -   2—J. F. CARDOSO A. SOULOUMIAC—December 1993—Blind beamforming for     non-gaussian signals IEE Proceedings-F, Vol. 140, No. 6, pp. 362-370 -   3—P. COMON—April 1994—Independent Component Analysis, a new concept     Elsevier—Signal Processing, Vol 36, no. 3, pp 287-314 

1. A method for localizing one or more sources, the sources being in motion relative to a network of sensors, the method comprising the following steps: separating the sources in order to identify the direction vectors associated with the response of the sensors to a source at a given incidence; associating the direction vectors a_(1p(1)m) . . . a_((K)m) obtained for the m^(th) source and, respectively, for the instants t₁ . . . t_(K) and for the wavelengths λ_(p(1)) . . . , λ_(p(K)), and localizing the m^(th) source from the components of the vectors a_(1p(1)m) . . . a_((K)m); wherein, in considering that, for each pair, (t_(k),λ_(p(k))), M_(k) vectors a_(kp(k)j) (1<j<M_(k)) have been identified, where M_(k) is the number of sources at k, the step of association for K pairs (t_(k),λ_(p(k))) comprises the following steps: 1) resetting of the process at k=0, m=1 and number of sources M=1 and, for all triplets (t_(k),λ_(p(k)),j), resetting a flag of association with a source flag_(kj) at flag_(kj)=0; 2) searching for an index j and a pair (t_(k),λ_(p(k))) such that the flag, flag_(kj)=0; 3) for first triplet (t_(k),λ_(p(k)),j) obtained at the step 2, effecting flag_(kj)=1 and resetting a test flag of association with the source of the first triplet link_(k′i)=0 for k′≠k and i′≠j and ind_(m)={k} and Φ_(m)={a_(kp(k)j)}; 4) determining the pair (t_(k′),λ_(p(k′))) minimizing a distance d_(kk) between pairs (t_(k′),λ_(p(k′))) and (t_(k),λ_(p(k))) such that kεind_(m) in the time-frequency space and in which there exists at least one vector b_(i)=a_(k′p(k′)i) such that flag_(k′i)=0 and link_(k′i)=0; 5) in using the relationship (4) defined here above, determining the index i(m) that minimizes the difference between the vectors a_(kp(k)m) such that kεind_(m) and the vectors b_(i) identified with the instant-wavelength pairs (t_(k′),λ_(p(k′))) for (1<i<M_(k′)) and flag_(k′i)=0 and link_(k′i)=0; 6) setting link_(k′i(m))=1: the test of association has been performed; 7) if d(a_(kp(k)m), b_(i(m)))<η and |t_(k)−t_(k′)|<Δt_(max) then: Φ_(m)={Φ_(m) b_(i(m))}, ind_(m)={ind_(m)k′}, flag_(k′i(m))=1; 8) if there is at least one doublet (t_(k′),λ_(p(k′))) and one index i such that link_(k′i)=0, then repeating the steps from the step 4; 9) defining the family of vectors Φ_(m)={a_(1p(1)m) . . . a_(K(m),p(K(m)),m)} associated with the source indexed by m in writing K(m)=cardinal(Φ_(m)); and 10) from the family of vectors Φ_(m)={a_(1p(1),m) . . . a_(K(M),p(K(M)),m)}, extracting the J instants t_(i)εind_(J)⊂ind_(m) which correspond to aberrant points located outside a defined zone; 11) returning to the step 3) if there is at least one triplet (t_(k),λ_(p(k)),j) such that flag_(kj)=0.
 2. The method according to claim 1 wherein the localizing step comprises the following steps: maximizing a criterion of standardized vector correlation L_(K)(x,y,z) in the position space (x,y,z) of a source with ${L_{K}\left( {x,y,z} \right)} = {\frac{{{b_{K}^{H}{v_{K}\left( {x,y,z} \right)}}}^{2}}{\left( {b_{K}^{H}b_{K}} \right)\left( {{v_{K}\left( {x,y,z} \right)}^{H}{v_{K}\left( {x,y,z} \right)}} \right)}\mspace{14mu}{with}}$ $\begin{matrix} {b_{K} = \begin{bmatrix} b_{1m} \\ \vdots \\ b_{Km} \end{bmatrix}} \\ {{= {{v_{K}\left( {x_{m},y_{m},z_{m}} \right)} + w_{K}}},{v_{K}\left( {x,y,z} \right)}} \\ {= {\begin{bmatrix} {b\left( {t_{1},\lambda_{p{(1)}},x,y,z} \right)} \\ \vdots \\ {b\left( {t_{K},\lambda_{p{(K)}},x,y,z} \right)} \end{bmatrix}\mspace{14mu}{and}}} \\ {w_{K} = \begin{bmatrix} w_{1m} \\ \vdots \\ w_{Km} \end{bmatrix}} \end{matrix}$ where W_(k) is the noise vector for all the positions (x, y, z) of a source.
 3. The method according to claim 2, wherein the vector b_(K) comprises a noise-representing vector whose components are functions of the components of the vectors a_(1m) . . . a_(Km).
 4. The method according to claim 3, comprising a step to compare the maximum values with a threshold value.
 5. The method according to claim 3, wherein the transmitters to be localized are mobile and wherein the vector considered is parametrized by the position of the source to be localized and the speed vector.
 6. A method according to claim 2, comprising a step in which the matrix of covariance R=E[W_(K)W_(K) ^(H)] of the noise vector is determined, and wherein the following criterion is maximized ${L_{K}^{\prime}\left( {x,y,z} \right)} = {\frac{{{b_{K}^{H}R^{- 1}{v_{K}\left( {x,y,z} \right)}}}^{2}}{\left( {b_{K}^{H}R^{- 1}b_{K}} \right)\left( {{v_{K}\left( {x,y,z} \right)}^{H}R^{- 1}{v_{K}\left( {x,y,z} \right)}} \right)}.}$
 7. The method according to claim 6, wherein the assessment of the criterion L_(K)(x,y,z) and/or the criterion L_(K′)(x,y,z) is recursive.
 8. The method according to claim 7, comprising a step to compare the maximum values with a threshold value.
 9. The method according to claim 7, wherein the transmitters to be localized are mobile and wherein the vector considered is parametrized by the position of the source to be localized and the speed vector.
 10. The method according to claim 6, comprising a step to compare the maximum values with a threshold value.
 11. The method according to claim 6, wherein the transmitters to be localized are mobile and wherein the vector considered is parametrized by the position of the source to be localized and the speed vector.
 12. The method according to claim 2, comprising a step to compare the maximum values with a threshold value.
 13. The method according to claim 2, wherein the transmitters to be localized are mobile and wherein the vector considered is parametrized by the position of the source to be localized and the speed vector.
 14. The method according to claim 1, comprising a step to compare the maximum values with a threshold value.
 15. The method according to claim 14, wherein the transmitters to be localized are mobile and wherein the vector considered is parametrized by the position of the transmitter to be localized and the speed vector.
 16. The method according to claim 1, wherein the transmitters to be localized are mobile and wherein the vector considered is parametrized by the position of the source to be localized and the speed vector.
 17. The method according to claim 1, comprising a step to compare the maximum values with a threshold value.
 18. The method according to claim 1, wherein the transmitters to be localized are mobile and wherein the vector considered is parametrized by the position of the source to be localized and the speed vector. 